# Laplace transform of Bessel functions

• Jan 22nd 2010, 08:20 AM
Laplace transform of Bessel functions
Dear all,

I would like to ask if anyone of you knows where to get the Laplace transform of the following function:

f(t) = I_n(a*t) * J_m(b*sqrt{t})

With

I_n() = Modified Bessel function of the first kind, of integer order n
J_m() = Bessel function of the first kind, of integer order m
a,b = real numbers

I was trying to look up in the book

A.P. Prudnikov, "Integrals and series, vol. 4 Laplace transforms"

but is not available in my library until march 8th! (Crying)

• Jan 22nd 2010, 11:58 AM
CaptainBlack
Quote:

Dear all,

I would like to ask if anyone of you knows where to get the Laplace transform of the following function:

f(t) = I_n(a*t) * J_m(b*sqrt{t})

With

I_n() = Modified Bessel function of the first kind, of integer order n
J_m() = Bessel function of the first kind, of integer order m
a,b = real numbers

I was trying to look up in the book

A.P. Prudnikov, "Integrals and series, vol. 4 Laplace transforms"

but is not available in my library until march 8th! (Crying)

I don't know the legal status of Library Genesis but they have a copy of volume 4 of Prudnikov in djvu format (for which you can download a free reader).

So google for Library Genesis and take it from there.

CB
• Jan 23rd 2010, 01:00 AM
Quote:

Originally Posted by CaptainBlack
I don't know the legal status of Library Genesis but they have a copy of volume 4 of Prudnikov in djvu format (for which you can download a free reader).

So google for Library Genesis and take it from there.

CB

Hello everybody,

I first thank CaptainBlack so much for answering, I even did not know about Library Genesis. I think I will use it a lot in the future!

However, Prudnikov's book does not give the result I am looking for. Any other suggestion?

• Jan 23rd 2010, 05:00 AM
shawsend
Quote:

. Any other suggestion?

Yeah, numerically. Ok, suppose I wanted to know what the Laplace Transform of that was at $\displaystyle n=2, m=3, a=1,b=1/2,s=2$ then I'd do the following in Mathematia:
In[3]:= n = 2; m = 3; a = 1; b = 1/2; s = 2; NIntegrate[  BesselI[n, a t] BesselJ[m, b Sqrt[t]] Exp[-s t],  {t, 0, \[Infinity]}] Out[8]= 0.000288939
$\displaystyle \mathcal{L}\left\{I_2(t) J_3(1/2\sqrt{t})\right\}\biggr|_{s=2}\approx 0.00029$
and if I needed to, I could do this for any values of the parameters and as long as the integration is well-behaved, I could obtain the value of the transform pretty accurately and if necessary, run a fit on the data points to obtain a least-square fit function $\displaystyle f(s)$ even if $\displaystyle s$ were complex.